Abstract

By means of a surface-integral formalism we derive the integral equations for diffuse photon density waves with boundary conditions corresponding to a diffuse–diffuse interface with index mismatch and solve them numerically without any approximation. These numerical results are verified with Monte Carlo simulations for the planar interface case. Since the application of the boundary condition to index-mismatched media is difficult, an approximation that yields very accurate values is found. This approximation can be easily introduced into analytical models, and a study of its limit of validity is shown. We demonstrate with numerical results that the multiple-scattering contribution that is due to surface roughness can be neglected, even when index mismatch is present.

Comparison of the normalized quantity |r-rsource||U(r)| calculated by the integral equations (solid curve) and by Monte Carlo simulation (circles) for the following parameters: medium 0, breast parameters μs0=75cm-1,g=0.8,μa0=0.035cm-1,n1=1.333; medium 1, breast tumor parameters μs1=50cm-1,g=0.8,μa0=0.24cm-1,n1=1.0. 40,000 photons were launched, and 20,000 interactions were permitted per photon in Monte Carlo. The dc (ω=0) source was at rsource=(0,1cm). The scan was performed in the z direction at x=0.

Comparison of the normalized quantity |r-rsource||U(r)| calculated by the integral equations (solid curve) and by Monte Carlo simulation (circles) for the following parameters: medium 0, breast parameters μs0=75cm-1,g=0.8,μa0=0.035cm-1,n1=1.333; medium 1, breast tumor parameters μs1=50cm-1,g=0.8,μa0=0.24cm-1,n1=2.0. 40,000 photons were launched, and 20,000 interactions were permitted per photon in Monte Carlo. The dc (ω=0) source was at rsource=(0,1cm). The scan was performed in the z direction at x=0.

Comparison of the wave amplitude |U(r)| calculated by the integral equations at zdetect=0.2cm (solid curves) and zdetect=-0.2cm (dotted curves) and by Monte Carlo simulation at zdetect=0.2cm (solid circles) and zdetect=-0.2cm (open circles) for two values of n1: (a) n0=1.333,n1=1.0; (b) n0=1.333,n1=3.0. Medium 0, breast parameters μs0=75cm-1,g=0.8,μa0=0.035cm-1; medium 1, breast tumor parameters μs1=50cm-1,g=0.8,μa0=0.24cm-1. 40,000 photons were launched, and 20,000 interactions were permitted per photon in Monte Carlo. The dc (ω=0) source was at rsource=(0,1cm). The scan was performed in the x direction at a constant z distance from the surface.

(a) Discretized curved surface studied to find the minimum value of Δ required for the expressions of RU,Jj→k to remain valid when a numerical calculation is performed when a surface with curvature radius R by elements dS is sampled. nˆ is the surface normal. (b) Detail of (a), where α=Δ/2.J-,+ represent the downward and upward flux densities, respectively.